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Find $x$ such that $$\sqrt{x+\sqrt{x+7}}\in \mathbb{N}$$

I tried many ways: $$\sqrt{x+\sqrt{x+7}}=n$$ $$\sqrt{x+\sqrt{x+7}}^2=n^2$$ $$x+\sqrt{x+7}=n^2$$

then solve for $x$ but didn't do with success.

I think this is the most difficult problem in my lifetime

Also $x$ must be made of $2$ digit.

Thanks to everybody for helping me understand this problem and its solution!

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There should be some other assumption in the problem, otherwise it's trivial that for any sufficiently large $n\in\mathbb{N}$ there is a solution to $\sqrt{x+\sqrt{x+7}}=n$ (and by sufficiently large, I mean $n>2$), and it is found just by isolating $x$ and resolving a quadratic polynomial. –  rewritten Dec 4 '13 at 10:53
    
An additional information must be needed.with above information we can find infinite values of x.we can understand that by plotting graph of $y$ vs $x$ where $y=x+\sqrt {x+7}$ and find corresponding value of x when y become perfect squares,by looking graph we can see that infinite values for $x$ is possible –  Jibin Joy K Dec 4 '13 at 11:44
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Is 42 the answer?! –  zerosofthezeta Dec 5 '13 at 5:56
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@zeta, that's the ultimate answer! –  tenpercent Dec 5 '13 at 6:36
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8 Answers 8

up vote 44 down vote accepted

If you just want to find some $x$, not all, you can try to find $x$ such that $\sqrt{x+7} = 7$, as then $$ \sqrt{x + \sqrt{x+7}} = \sqrt{x+7} = 7 \in \mathbb N.$$ Can you see such an $x$?

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MARVELLOUS!!! THANK YOU SO MUCH!!! –  Geirg Von Kirk Dec 4 '13 at 13:08
    
But i have problem how you derived ths equationn $\sqrt{x + \sqrt{x+7}} = \sqrt{x+7}$ –  Geirg Von Kirk Dec 4 '13 at 13:13
    
@9abab You need $\sqrt{x+7} = 7$. When you do that, you can use $\sqrt{x + \sqrt{x+7}} = \sqrt{x+7}$. Can you find an $x$ so $\sqrt{x+7} = 7$? –  kba Dec 4 '13 at 13:31
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You deserve a money price :) –  Geirg Von Kirk Dec 4 '13 at 19:11
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Coincides with the answer to the Ultimate Question of Life, the Universe, and Everything –  Alex Dec 7 '13 at 17:44
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$$x + 7 = ((n^2)-x)^2 = n^4 - 2n^2x + x^2$$ $$x^2-(2n^2+1)x+n^4-7=0$$ $$x_{1,2}=\frac{2n^2+1 \pm \sqrt{(2n^2+1)^2-4(n^4+7)}}{2}\\ =\frac{2n^2+1 \pm \sqrt{4n^4+4n^2+1-4n^4+28}}{2}\\ =\frac{2n^2+1 \pm \sqrt{4n^2+29}}{2}\\$$

This is an integer if and only if $ \sqrt{4n^2+29}$ is an integer (the converse follows immediately from the observation that if $\sqrt{4n^2+29}$ is an integer, it must be odd.

Claim 1: If $\sqrt{4n^2+29}$ is an integer, then $n \leq 7$.

Proof: $$\sqrt{4n^2+29} > \sqrt{4n^2}=2n \,.$$ Thus $$\sqrt{4n^2+29} \geq 2n+1 \,.$$ Hence $$4n^2+29 \geq 4n^2+4n+1 \,.$$

This implies that $n \leq 7$, with equality if and only if $n=7$.

Claim 2: If $\sqrt{4n^2+29}$ is an integer, and $n \neq 7$ then $n \leq 1$.

Proof: Exactly like in Claim $1$ $$\sqrt{4n^2+29} \geq 2n+1 \,.$$

But we proved that we only have equality for $n=7$. Thus $$\sqrt{4n^2+29} > 2n+1 \,.$$

As $\sqrt{4n^2+29}$ is odd, we get $$\sqrt{4n^2+29} \geq 2n+3 \,.$$ $$4n^2+29 \geq 4n^2+12n+9 \,.$$ $$29 \geq 12n+9 \,.$$

Thus $n \leq 1$.

Thus we showed that the only $n$ which can work are $n = 0, n = 1$ and $n = 7$.

If $$n=0 \Rightarrow x_{1,2} =\frac{1 \pm \sqrt{29}}{2} \notin \mathbb Z$$ $$n=1 \Rightarrow x_{1,2} =\frac{2+1 \pm \sqrt{4+29}}{2}\notin \mathbb Z$$ $$n=7 \Rightarrow x_{1,2} =\frac{99 \pm 15}{2}$$

P.S. Don't forget that we squared couple times, which means that the solutions we got are solutions to $$x + 7 = ((n^2)-x)^2 = n^4 - 2n^2x + x^2$$ but not necessarily to the original question.

One needs to check them in the original equation, and only one works.

The reason why the other solution doesn't work is because it appears as an extra solution when we square: $$\sqrt{x + 7 }= (n^2)-x$$

If we observe in the original equation that $n^2 \geq x$, this eliminates the wrong extra solution .

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+1 for the only complete answer :) –  nbubis Dec 4 '13 at 22:02
    
last value is for n=7, btw (gives the solution given by martini, too) –  njzk2 Dec 4 '13 at 23:24
    
It is easy to show that if $x\in\mathbb{R}$ such that $\sqrt{x+\sqrt{x+7}}=n\in\mathbb{N}$ then $\displaystyle \frac{1-\sqrt{29}}{2}\le x\le n^2$ must hold. Therefore we cannot have solutions of the form $\displaystyle\frac{2n^2+1+\sqrt{4n^2+29}}{2}$. The only valid ones that yield $n\in\mathbb{N}$ are of the form $\displaystyle\frac{2n^2+1-\sqrt{4n^2+29}}{2}$. –  tcmtan Dec 5 '13 at 2:15
    
you see, $\frac{99+15}{2} = 57$, and $\sqrt{57 + \sqrt{57 + 7}} = \sqrt{57 + 8} = \sqrt{65}$. It happens. –  tenpercent Dec 5 '13 at 5:09
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@N.S. If I wasn't clear then I meant the same thing you wrote. You should then clarify that only solutions of the form $\displaystyle\frac{2n^2+1-\sqrt{4n^2+29}}{2}$ satisfy the original equation. –  tcmtan Dec 5 '13 at 5:38
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$x + 7 = ((n^2)-x)^2 = n^4 - 2n^2x + x^2$ which is quadratic equation considering x. Can you go further?

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This leads to $(x-n^2)^2=x+7$. Where does one go now? –  Jp McCarthy Dec 4 '13 at 13:39
    
@JpMcCarthy, wrong way! $x = \frac{(2n^2 + 1)\pm \sqrt{4n^4+4n^2+1+28-4n^4 }}{2}$, but it seems that my guess was also wrong, as I can't imagine what to do with the thing under the root. I only know that it should be a square of some odd integer. Sorry, guys. Besides, you have a nice solution up there. –  tenpercent Dec 4 '13 at 13:52
    
@JpMcCarthy To be more accurate, $(2n)^2 + 29 = k^2$ implies that $2n \leq 29 $ and $n \leq 14$. There are only 14 such positive integers that $n \leq 14$, so we can check if any of them is the solution to the problem. Or we could try to solve $4n^2 + 1 = k^2 \mod 14 $, which leaves us less variants to consider. –  tenpercent Dec 4 '13 at 14:17
    
Begs the question why did this get devoted? –  Jp McCarthy Dec 4 '13 at 15:33
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@JpMcCarthy, because the upvoted hint is awesome. –  tenpercent Dec 4 '13 at 16:06
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Trial and Error

I assume you want $x$ to be an integer. You might want $x+7$ to be a square so that $\sqrt{x+7}\in\mathbb{N}$... $x+7\in\{1,4,9,16,25,36,49,64\dots\}$ and hence $$\begin{align}x&\in\{-6,-3,2,18,29,42,57,\dots\} \end{align}$$

But you want $x+\sqrt{x+7}$ to be a square so look at the corresponding values of it:

$$x+\sqrt{x+7}\in\{-5,-1,5,22,34,49,65,\dots\}.$$

One of these is squares, corresponding to $x=42$.

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thanks this is also very helpful!!! –  Geirg Von Kirk Dec 4 '13 at 19:48
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42 is the solution. Why I am not surprised? –  Avidanborisov Dec 4 '13 at 20:20
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$x = -3$ gives $x + \sqrt{x + 7} = -1$, not 1. –  hunse Dec 5 '13 at 16:14
    
@hunse Thank you. Edited. –  Jp McCarthy Dec 5 '13 at 16:51
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We want to find $a \in \mathbb{N}$ such that $$a = \sqrt{x + \sqrt{x + 7}}.$$ Also implicit in the question was that $x \in \mathbb{N}$, but that wasn't stated, only implied by the statement that $x$ is two-digit.

Let $n = \sqrt{x + 7}$. Then $x = n^2 - 7$. Substituting into the original equation: $$\begin{align} a = & \sqrt{n^2 + n - 7}\\ 0 = & n^2 + n - 7 - a^2 \end{align}$$ Solving for $n$: $$n = -\frac{1}{2} \pm \frac{1}{2}\sqrt{4 a^2 + 29}$$ Plugging any $a > 1$ into this equation will yield a valid value for $n$ and therefore $x$, but they will generally be non-integer.

For $n$ and therefore $x$ to be integer, $4 a^2 + 29$ must be a perfect square, and furthermore an odd perfect square. We can rewrite this as $(2a)^2 + 29 = b^2$, where $a, b \in \mathbb{N}$, which means two perfect squares must have a difference of 29, and this only occurs for $(2a)^2 = 14^2$ and $b^2 = 15^2$. Therefore, solving backwards, $a = 7$, $n = 7$, and $x = 42$.

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Well, $x=42$ would be more true. ;-) –  egreg Dec 4 '13 at 21:48
    
@egreg: Thanks! –  hunse Dec 4 '13 at 23:19
    
@RossMillikan: $x = -3$ results in $n = 2$ and $a = \sqrt{-1}$, which does satisfy $0 = n^2 + n - 7 - a^2$, but breaks the restriction that $a$ is a natural number (positive integer). Thanks for pointing it out, though. –  hunse Dec 4 '13 at 23:23
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To see what's going on in this problem, it helps to generalize it slightly. (This should also help put martini's wonderful answer in context.) Let's look for integers $x$ that make an expression of the form

$$\sqrt{x+\sqrt{x+A}}\in\mathbb{N}$$

for an arbitrary $A\in\mathbb{N}$.

As other answers have shown, setting the expression in question equal to $k$ leads to the quadratic equation

$$x^2-(2k^2+1)x+(k^4-A)=0$$

for which the discriminant is

$$\Delta = (2k^2+1)^2-4(k^4-A)=4k^2+4A+1$$

For $x$ to be an integer, the discrimant must be a square, say $\Delta = n^2$. This implies

$$(n+2k)(n-2k)=4A+1$$

Now any factorization of the integer $4A+1$, say $4A+1=ab$, gives integer values for $n$ and $k$ (with non-negative $k$ if we take $a\ge b$): Setting $(n+2k)=a$ and $(n-2k)=b$ implies

$$n={a+b\over2}\quad\text{and}\quad k={a-b\over4}$$

(The denominators here, especially the $4$, may appear problematic, but note that $4A+1$ is congruent to $1$ mod $4$, hence its factors must be odd numbers that are both congruent to either $1$ or $3$ mod $4$.) Each factorization gives two values for $x$:

$$x={2k^2+1-n\over2}\quad\text{and}\quad x={2k^2+1+n\over2}$$

However, only the first of these is a solution of $\sqrt{x+\sqrt{x+A}}=k$; the other is a solution of $\sqrt{x-\sqrt{x+A}}=k$.

In particular, we always have the factorization $a=4A+1$, $b=1$, and this gives $n=2A+1$, $k=A$, leading to $x=A^2-A$. This is the solution in martini's wonderful answer.

For $A=7$, the number $4A+1=29$ is prime, so this is the only factorization. But for $A=11$, for example, we wind up with three values of $x$ that make $\sqrt{x+\sqrt{x+11}}$ an integer: $x=110$, $x=5$, and $x=-2$, corresponding to the factorizations $45\cdot1$, $15\cdot3$, and $9\cdot5$.

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just to it by guesswork , x=42

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guesswork isn't good in mathematics –  Geirg Von Kirk Dec 4 '13 at 13:11
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@9abab Au contraire. Guesswork is very good in mathematics... you just have to be careful. If the question was to find all $x$ guesswork is not OK... how would you know there are not more solutions. Here we are only looking for one and 42 is fine. Perhaps Sahil can explain how $x=42$ was guessed... by a trial and error. –  Jp McCarthy Dec 4 '13 at 13:22
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Quite some mathematical problems came to a solution when someone started guessing. –  drhab Dec 4 '13 at 13:42
    
The difference here is between random guesses and educated guesses. Random guesses are rarely good. But educated guesses can help a lot, but require a thorough understanding, experience and intuition. –  Turion Dec 4 '13 at 14:07
    
Wrong again. Random guesses is precisely the basis of a revolution that has happened in mathematics that has solved many problems already. Educated guesses are just an accident, getting lucky of having a problem so simple that it allows some simplification, a rarity in the sea of all mathematical problems. –  ABC Dec 4 '13 at 15:56
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You're mentioning that $x$ should have two digits. Write a small program that runs through all integers from 10 to 99, computes your expression and finds that 42 is the only solution.

Example in Python:

>>> import math
>>> for x in range(10,100):
...     n = math.sqrt(x + math.sqrt(x + 7))
...     if float(int(n)) == n:
...             print("The solution is {0}".format(x))
...
The solution is 42
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Pragmatic, but probably not what Geirg's teacher is looking for... –  fluffy Dec 5 '13 at 6:59
    
@fluffy, what makes you think so? I think I'm the only one who used the fact that $x$ should have two digits, and it's a legitimate information to use. –  Turion Dec 5 '13 at 12:43
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